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Групповые структуры на алгебраических многообразиях
Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика). 2021. Т. 500. № 1. С. 52-54.
It is explored to which extent the group variety of an algebraic group determines its group structure.
Popov V., Известия РАН. Серия математическая 2022 Т. 86 № 5 С. 73-96
We explore to what extent the group variety of a connected algebraic group or the group manifold of a real Lie group determines its group structure. ...
Added: June 9, 2022
Ivan V. Arzhantsev, Yulia I. Zaitseva, Kirill V. Shakhmatov, Proceedings of the Steklov Institute of Mathematics 2022 Vol. 318 No. 1 P. 13-25
Let X be an algebraic variety such that the group Aut(X) acts on X transitively. We define the transitivity degree of X as the maximum number m such that the action of Aut(X) on X is m-transitive. If the action of Aut(X) is m-transitive for all m, the transitivity degree is infinite. We compute the transitivity degree for all quasi-affine toric varieties and for many homogeneous spaces of algebraic groups. We also discuss a conjecture and ...
Added: November 5, 2022
Arzhantsev I., Zaitseva Y., Shakhmatov K., Труды Математического института им. В.А. Стеклова РАН 2022 Т. 318 С. 17-30
Let X be an algebraic variety such that the group Aut(X) acts on X transitively. We define the transitivity degree of X as a maximal number m such that the action of Aut(X) on X is m-transitive. If the action of Aut(X) is m-transitive for all m, the transitivity degree is infinite. We compute the transitivity degree for all quasi-affine toric varieties and for many homogeneous spaces of algebraic groups. Also we discuss a ...
Added: November 4, 2022
Beldiev I., Results in Mathematics 2023 Vol. 78 No. 5 Article 192
We study induced additive actions on projective hypersurfaces, i.e. effective regular actions of the algebraic group G_a^m with an open orbit that can be extended to a regular action on the ambient projective space. We prove that if a projective hypersurface admits an induced additive action, then it is unique if and only if the hypersurface is ...
Added: August 7, 2023
Vladimir L. Popov, / Cornell University. Series math "arxiv.org". 2021. No. 2105.12861.
Starting with exploration of the possibility to present the underlying variety of an affine algebraic group in the form of a product of some algebraic varieties, we then explore the naturally arising problem as to what extent the group variety of an algebraic group determines its group structure. ...
Added: May 28, 2021
Р.С. Авдеев, Труды Московского математического общества 2010 Т. 71 С. 235-269
A spherical homogeneous space G/H of a connected semisimple algebraic group G is called excellent if it is quasi-affine and its weight semigroup is generated by disjoint linear combinations of the fundamental weights of the group G. All the excellent affine spherical homogeneous spaces are classified up to isomorphism. ...
Added: February 25, 2014
Vladimir L. Popov, European Journal of Mathematics 2016 Vol. 2 No. 1 P. 283-290
According to the classical theorem, every algebraic variety
endowed with a nontrivial rational action of a connected linear algebraic
group is birationally isomorphic to a product of another algebraic variety
and the projective space of a positive dimension. We show that the classical proof of this theorem
actually works only in characteristic 0 and we give a characteristic free
proof ...
Added: February 2, 2016
Группы базовых автоморфизмов картановых слоений моделируемых на неэффективных картановых геометриях.
Zhukova N., Sheina K., Труды Математического центра им. Н.И. Лобачевского 2015 Т. 52 С. 73-74
Исследуются картановы слоения, то есть слоения допускающие трансверсальную картанову геометрию. Рассматривается общая ситуация, когда картанова геометрия может быть неэффективной. Найдено достаточное условие для того, чтобы полная группа базовых автоморфизмов картанова слоения со связностью Эресмана допускала единственную структуру конечномерной группы Ли в категории картановых слоений, где изоморфизмы сохраняют как слоение, так и трансверсальную геометрию. Получены некоторые ...
Added: October 14, 2015
Р.С. Авдеев, Труды Московского математического общества 2011 Т. 72 № 1 С. 5-62
We develop a structure theory of connected solvable spherical subgroups in semisimple algebraic groups. Based on this theory, we obtain an explicit classification of all such subgroups up to conjugacy. ...
Added: February 25, 2014
Р.С. Авдеев, Математические заметки 2013 Т. 94 № 1 С. 22-35
For an arbitrary connected solvable spherical subgroup H of a connected semisimple algebraic group G, we compute the group N_G(H), the normalizer of H in G. Thereby we complete a classification of all (not necessarily connected) solvable spherical subgroups in semisimple algebraic groups. ...
Added: February 25, 2014
Arzhantsev I., Zaitseva Y., Russian Mathematical Surveys 2022 Vol. 77 No. 4 P. 571-650
We survey recent results on open embeddings of the affine space C^n into a complete algebraic variety X such that the action of the vector group G_a^n on C^n by translations extends to an action of G_a^n on X. We begin with the Hassett–Tschinkel correspondence describing equivariant embeddings of \mathbb{C}^n into projective spaces and present its generalization for embeddings into projective hypersurfaces. Further sections deal with embeddings into flag ...
Added: February 26, 2023
V. L. Popov, Mathematical notes 2018 Vol. 103 No. 5 P. 811-819
We prove that the family of all connected n-dimensional real Lie groups is uniformly Jordan for every n. This
implies that all algebraic (not necessarily affine) groups over fields of characteristic zero and some
transformation groups of complex spaces and Riemannian manifolds are Jordan. ...
Added: April 13, 2018
Arzhantsev I., Алгебра и анализ 2022 Т. 34 № 2 С. 1-55
В работе дан обзор результатов последних лет о кратной транзитивности действий групп автоморфизмов аффинных алгебраических многообразий. Рассматривается свойство бесконечной транзитивности действия группы специальных автоморфизмов и эквивалентное ему свойство гибкости многообразия. Данные свойства имеют важные алгебраические и геометрические следствия, и при этом они выполнены для широких классов многообразий. Отдельно изучаются случаи, когда бесконечная транзитивность имеет место ...
Added: March 14, 2022
Shirokov D., Advances in Applied Clifford Algebras 2010 Vol. 20 No. 2 P. 411-425
In this paper we present new formulas, which represent commutators and anticommutators of Clifford algebra elements as sums of elements of different ranks. Using these formulas we consider subalgebras of Lie algebras of pseudo-unitary groups. Our main techniques are Clifford algebras. We have found 12 types of subalgebras of Lie algebras of pseudo-unitary groups. ...
Added: June 16, 2015
Arzhantsev I., Liendo A., Stasyuk T., Journal of Pure and Applied Algebra 2021 Vol. 225 No. 2 P. 106499
Let X be a normal variety endowed with an algebraic torus action. An additive group action alpha on X is called vertical if a general orbit of alpha is contained in the closure of an orbit of the torus action and the image of the torus normalizes the image of alpha in Aut(X). Our first result in this paper ...
Added: July 29, 2020
Roman Avdeev, Petukhov A., Transformation Groups 2021 Vol. 26 No. 3 P. 719-774
Let G be a symplectic or special orthogonal group, let H be a connected reductive subgroup of G, and let X be a flag variety of G. We classify all triples (G, H, X) such that the natural action of H on X is spherical. For each of these triples, we determine the restrictions to ...
Added: September 2, 2020
Sheina K., Известия высших учебных заведений. Поволжский регион. Физико-математические науки 2021 Т. 1 № 1 С. 49-65
The basic automorphism group of a Cartan foliation (M,F) is the quotient group of the automorphism group of (M, F ) by the normal subgroup, which preserves every leaf invariant. For Cartan foliations covered by fibrations, we find sufficient conditions for the existence of a structure of a finite-dimensional Lie group in their basic automorphism ...
Added: December 16, 2020
Roman Avdeev, Transformation Groups 2021 Vol. 26 No. 2 P. 403-431
Given a connected reductive complex algebraic group G and a spherical subgroup H⊂G, the extended weight monoid Λˆ+G(G/H) encodes the G-module structures on spaces of regular sections of all G-linearized line bundles on G/H. Assuming that G is semisimple and simply connected and H is specified by a regular embedding in a parabolic subgroup P⊂G, ...
Added: September 9, 2021
Roman Avdeev, Cupit-Foutou S., Advances in Mathematics 2018 Vol. 328 P. 1299-1352
Given a connected reductive algebraic group G and a finitely generated monoid Γ of dominant weights of G, in 2005 Alexeev and Brion constructed a moduli scheme M_Γ for multiplicity-free affine G-varieties with weight monoid Γ. This scheme is equipped with an action of an `adjoint torus' T_ad and has a distinguished T_ad-fixed point X_0. ...
Added: February 25, 2018
V. L. Popov, Transformation Groups 2011 Vol. 16 No. 3 P. 827-856
Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a
closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove
that in arbitrary G such a cross-section exists if and only if the ...
Added: March 16, 2013
Sheina K., / Cornell University. Series arXiv "math". 2020. No. 04348v1.
The basic automorphism group of a Cartan foliation (M, F) is the quotient group of the automorphism group of (M, F) by the normal subgroup, which preserves every leaf invariant. For Cartan foliations covered by fibrations, we find sufficient conditions for the existence of a structure of a finite-dimensional Lie group in their basic automorphism groups. Estimates ...
Added: December 9, 2020
Arzhantsev I., Celik D., Hausen J., Journal of Algebra 2013 Vol. 387 P. 87-98
Given an action of an affine algebraic group with only trivial characters on a factorial variety, we ask for categorical quotients. We characterize existence in the category of algebraic varieties. Moreover, allowing constructible sets as quotients, we obtain a more general existence result, which, for example, settles the case of afinitely generated algebra of invariants. ...
Added: November 13, 2013
Р.С. Авдеев, Петухов А. В., Математический сборник 2014 Т. 205 № 9 С. 3-48
For every finite-dimensional vector space V and every V-flag variety X we list all connected reductive subgroups in GL(V) acting spherically on X. ...
Added: October 22, 2014
Попов В. Л., Известия РАН. Серия математическая 2019 Т. 84 № 4 С. 194-225
The rst group of results of this paper concerns the compressibility of finite subgroups of the Cremona groups. The second concerns the embeddability of other groups in the Cremona groups and, conversely, the Cremona groups in
other groups. The third concerns the connectedness of the Cremona groups. ...
Added: July 31, 2019